SIAM Review, Volume 65, Issue 1, Page 259-260, February 2023.
In this issue the Education section presents two contributions. The first paper is “Chaos Game Representation” (CGR), written by Eunice Y. S. Chan and Robert M. Corless. The chaos game is an algorithm which allows us to produce pictures like fractal structures associated with one-dimensional sequences of integers and, in this way, to “visualize” such sequences. In particular, these pictures can help to generate conjectures and give an intuition of the “distance” between sequences. CGR is based on an algorithm from chaotic dynamics called the “chaos game,” popularized by Barnsley. It involves a polygon, a dividing rate, and randomly chosen sequences of vertices of the polygon leading to various fractal-like patterns depending on the game board and the dividing rate. For a triangular game board and some choices of dividing rate, one gets the Sierpinski triangle. The appearance of patterns is explained by the overlapping of the attractors of CGR. This overlapping can be controlled by an appropriate choice of the dividing rate. The authors also demonstrate, in an experimental way, that the number of vertices in the game board may influence the final patterns. The article provides a nice description of how CGR has been applied to DNA and protein (amino acids) sequencing and visualization. Finally, the CGR is also discussed when the random choice of vertices is replaced by digits of $\pi$, Fibonacci sequence, and prime numbers, getting various patterns in each case. The construction of CGR given in this paper can be introduced in a “dynamical systems” or “modeling” courses classroom. This teaching module also aims to stimulate a discussion about randomness and its meaning. The second paper, “Surprises in a Classic Boundary-Layer Problem,” is presented by William A. Clark, Mario W. Gomes, Arnaldo Rodriguez-Gonzalez, Leo C. Stein, and Steven H. Strogatz. It provides a detailed analysis of a nonlinear boundary-value problem $$ (1.1) \varepsilon y ” = yy' - y , \quad y(0) = 1 ,\quad y(1) = -1 $$ and exhibits a number of previously overlooked properties of its solutions. Equation (1.1) is considered in the classic textbook by Mark Holmes [Introduction to Perturbation Methods, Springer, New York, 1995], where it is argued that it has a unique solution, however after imposing some convexity/concavity assumptions on solutions near the boundary. In the absence of such an assumption it turns out that (1.1) actually has three solutions whenever $\varepsilon >0$ is sufficiently small. In particular, the authors show the existence of a pitchfork bifurcation parameter $\varepsilon_c=0.2159869288903\dots$ such that for all $0 < \varepsilon < \varepsilon_c $ there are three solutions of the above equation, while for $\varepsilon \geq \varepsilon_c $ the solution is unique. The bifurcation parameter $\varepsilon_c $ is obtained by an explicit elegant calculation. Then the value of $y'(0)$ for each solution is discussed whenever $\varepsilon >0$ is sufficiently small. For two of them it is transcendentally close to 1, and explicit estimates of this closeness are explained. Then the occurring pitchfork bifurcation is discussed in more detail. The study relies on converting the second order ODE into a system of two first order ODEs and then investigating the phase portrait to understand qualitative properties of solutions. In particular, solutions are conservative and some of them enjoy symmetry-like properties. Finally, using the obtained estimates, the shooting method is applied to solve (1.1) numerically. The detailed proofs and calculations make this paper almost self-contained and provide a methodology of a complete analysis of (1.1). This problem could be included in courses on perturbation methods, applied dynamical systems, or numerical analysis.

中文翻译：

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教育

SIAM Review，第 65 卷，第 1 期，第 259-260 页，2023 年 2 月。
在本期中，教育部分提出了两项​​贡献。第一篇论文是“Chaos Game Representation”（CGR），由 Eunice YS Chan 和 Robert M. Corless 撰写。混沌游戏是一种算法，它允许我们生成与一维整数序列相关联的分形结构等图片，并以这种方式“可视化”此类序列。特别是，这些图片可以帮助产生猜想并给出序列之间“距离”的直觉。CGR 基于混沌动力学中的一种算法，称为“混沌博弈”，由巴恩斯利 (Barnsley) 推广。它涉及多边形、分割率和随机选择的多边形顶点序列，根据游戏板和分割率导致各种类似分形的图案。对于三角棋盘和分度率的一些选择，一个得到谢尔宾斯基三角形。图案的出现可以用 CGR 吸引子的重叠来解释。这种重叠可以通过适当的分隔率选择来控制。作者还以实验方式证明，游戏板上的顶点数量可能会影响最终模式。这篇文章很好地描述了 CGR 如何应用于 DNA 和蛋白质（氨基酸）测序和可视化。最后，还讨论了当顶点的随机选择被$\pi$、斐波那契数列和素数替换时的CGR，在每种情况下得到不同的模式。本文给出的 CGR 的构造可以在“动力系统”或“建模”课程课堂上引入。该教学模块还旨在激发关于随机性及其意义的讨论。第二篇论文“Surprises in a Classic Boundary-Layer Problem”由 William A. Clark、Mario W. Gomes、Arnaldo Rodriguez-Gonzalez、Leo C. Stein 和 Steven H. Strogatz 发表。它详细分析了非线性边值问题 $$ (1.1) \varepsilon y ” = yy' - y , \quad y(0) = 1 ,\quad y(1) = -1 $$ 并展示了其解决方案以前被忽视的属性的数量。方程式 (1.1) 在 Mark Holmes 的经典教科书 [Introduction to Perturbation Methods, Springer, New York, 1995] 中被认为具有唯一解，但是在对接近方程的解施加一些凸/凹假设之后边界。在没有这样的假设的情况下，只要 $\varepsilon >0$ 足够小，(1.1) 实际上就有三个解。尤其，作者证明了干草叉分岔参数 $\varepsilon_c=0.2159869288903\dots$ 的存在，使得对于所有 $0 < \varepsilon < \varepsilon_c $，上述方程有三个解，而对于 $\varepsilon \geq \varepsilon_c $，解是唯一的。分岔参数 $\varepsilon_c $ 是通过显式优雅计算获得的。然后，只要 $\varepsilon >0$ 足够小，就会讨论每个解的 $y'(0)$ 的值。对于其中两个，它超越地接近于 1，并解释了对这种接近度的明确估计。然后更详细地讨论了发生的干草叉分叉。该研究依赖于将二阶 ODE 转换为两个一阶 ODE 的系统，然后研究相图以了解解决方案的定性特性。尤其，解决方案是保守的，其中一些具有类似对称的特性。最后，使用获得的估计值，应用射击方法对（1.1）进行数值求解。详细的证明和计算使本文几乎自成体系，并提供了完整分析（1.1）的方法论。这个问题可以包含在微扰方法、应用动力系统或数值分析的课程中。